July 1968Ind. J. Physiol. & Pharmacal.

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Corson, The lung clinicalChicago (1954).

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adults. Jour. Ind. Med,

ction; effects of pulmonary

. urn Breathing Capacity19:20 1965

A SIMPLE CALCULATION FOR ERGOGRAPHY

ByP. SIMHADRI* AND P. SUOHIRt

Department of Physiology, Kurnool Medicul College, Kurnool.

In experiments using Mosso's ergograph, the calculation of the total work done involvesa long and tedious measurement of the ordinates on the kymograph record. It is possible, nodoubt, to use a mechanical counter fixed to the pulley or to the moving lever. This countergives the excursions of the recording lever as the total distance or the height to which the masshas been lifted. In the absence of such a mechanical device a simple formula to determine thetotal distance is suggested and then the work done can easily be calculated.

The following mathematical solution needs only four measurements and thus considerablysimplifies the calculations. Though an approximation this method is accurate enough fo.experimental purposes as well as for assessment of the progress in patients of myasthenia gravis.

If M is the mass used in the experiment, and if hI' h2' h3, hn are the heights of thesuccessive ordinates (displacements) the total work is given by

W = Mgh,+ Mgh2+Mgh~ Mgh,W=Mg (h1+h2+h3 hn)W=MgL'h

"'i'o....•....•

...•.

Fig. 1.Schematic reproduction of an ergographic tracing. The three lines drawn are :

(1) Hypotenuse touching all ordinates. (2) The base. (3) The first ordinate. The three measurements art(i) The angle between the hvpotensue and the base(ii) The height of the first ordinate (iii) The number of ordinates. This is derived by the total length

of the base i.e., time divided by intervals of ergographic recording.

·Present address: Professor of Physiology, Osmania Medical College, Hyderabad-l.A.P.tPre<ent address: Department of Physics, Madras Christian College, Madras-Ss

116 Simhadri and Sudhir June 1968Ind. J. Physiol. & Pharmacel

For calculation a 'mean' line is drawn through the ends of the ordinates as shown in thefigure 1. The ordinates touched by this line have successively decreasing heights the decrementbeing a constant. Let it be denoted by L,'h'.

It is obvious then that the successive heights constitute (numerically) an arthmetical pro-gression. Now in an arithmetical progression of 'n' numbers with a common difference 'd'and the number which comes first i.e., the initial number 'a', the sum of all numbers is givenb~

nS=2 2a+(n-l)dHence, in the problem considered if h is the intitial displacement and there are 'n' displacements.the sum of the heights of all the ordinates is :

n.Eh=T 2h1X(n-l) h (1)

• xTT

Fig.2.A graphic indication of the Successive decrements in the ordinates. Decrement (6) is calculated as follows:6h=X.TancP 'X' is derived by dividing the length of"T" by 'n'. Angle cPis angle between the hypotenuse andbas«.

hIn figure 2 it is seen that if the angle made by the mean line with the x-axis is </>,then ten -xwhere x is the intercept on the x-axis. Therefore h=x. tan</>Assuming 'x' to be constant,x.-! where T is the length of x-axis which represents the durationa and'n' the number of

nordinates, we get,

6h=J tan O.n

Substituting the value of L,h in equation (1) we get :n T.Eh=T 2h1+(n-l) n tans (2)

Since the progression is decreasing

L'h= ~ [2hl-(n-l) !tan,p l (3)

Hence total work done is given by

Wc=Mg ~ [ 2hl -(n-1) !tancP] ergs.

Note (a) If final displacement is zero, then tancP= ~

June 1968Ind. J. Physiol, & Pharmacol.

ordinatesas shown in theingheights the decrement

y) an arthmetical pro-a common difference 'd'of all numbers is given by

thereare 'n' displacements.

t (6) is calculated as follows:/RIfle between the hypotenuse and

•• .I.. h he x-ax1S1S'/', t en ten -xsuming'x' to be constant,and'n' the number of

Volume 12

Number 3

Ergography Calculation 117

WHENCE WE Obtain, W=Mgh1 O+n)12

(b) If final displacement is not zero, (then as shown in figure 3,W- Mgn 2hl -(n-l)y.

- 2 2

- y.tan</>-T' Whence,

IVI,

T

Fig. 3.

All incomplete Ergographic Record. The line drawn touching the tips of the ordinates does 1I0t meet the baseline as in Fig. 1. A parallel line is drawn to 'T', the distance being the last ordinate.

Explanatory note for (a) If final displacement is zero then tan</>=~ Substitutingthis value of tan </>in the final equation we get,

n} T hIW=Mg T 2h1 -(n-l)'n' T

_Mgn.hl (1+n) Mghdl+n)---2-' -n- 2.

EXPLANATION AND ELUCIDATION

The actual ergograph does show variations of the heights of the successive ordinates. Itis expected that we draw a line indicating the mean of the heights. It will be seen that there area number of ordinates above the line and a number of them below. When a straight line isdrawan touching the ordinates it automatically means that the decrements are uniform orconstant and the slope of the 'mean' line indicates the rate of decrement. Just as in other scat-tergrams a mean line inidcates the rate of change, here too it is assumed that the mean straightline indicates the decrements which are constant.

Actual comparison between the calculated results and mea ured results coincide to ahigh degree, hence the suggestion of this formula, which is quick and simple obviating thelabourous process of measuring each ordinate with a divider or a scale.

Present address : Professor of Physiology Osmania Medical College Hyderabad 1. A. P.